The PDE-ODI principle and cylindrical mean curvature flows

TL;DR

The authors develop the PDE-ODI principle, a general mechanism converting broad classes of parabolic PDEs arising in geometric flows into finite-dimensional ordinary differential inequalities for dominant modes. They apply this to ancient and asymptotically cylindrical mean curvature flows, obtaining high-order asymptotics and a sharp dichotomy between dominant linear and quadratic modes, encoded by a symmetric, nonnegative matrix invariant $\mathsf{Q}$. In the linear-mode case, they prove a complete classification: the flow is bowl soliton times a Euclidean factor, while the quadratic-mode case yields arbitrary polynomial asymptotics parameterized by $\mathsf{Q}$ and realized by ancient ovals. The framework unifies and rederives classical results (tangent-flow uniqueness and cylinder rigidity) without relying on Łojasiewicz-Simon-type inequalities and provides sharp control of ambient gauging via a gauged, rescaled MCF. The results pave the way toward a full classification of ancient, asymptotically cylindrical flows and establish a versatile toolkit (pseudolocality, gauging, high-order ODI analysis) for future geometric-flow problems.

Abstract

We introduce a new approach for analyzing ancient solutions and singularities of mean curvature flow that are locally modeled on a cylinder. Its key ingredient is a general mechanism, called the \emph{PDE--ODI principle}, which converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This principle bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. As an application, we establish the uniqueness of the bowl soliton times a Euclidean factor among ancient, cylindrical flows with dominant linear mode. This extends previous results on this problem to the most general setting and is made possible by the stronger asymptotic control provided by our analysis. In the other case, when the quadratic mode dominates, we obtain a complete asymptotic expansion to arbitrary polynomial order, which will form the basis for a subsequent paper. Our framework also recovers and unifies several classical results. In particular, we give new proofs of the uniqueness of tangent flows (due to Colding-Minicozzi) and the rigidity of cylinders among shrinkers (due to Colding-Ilmanen-Minicozzi) by reducing both problems to a single ordinary differential inequality, without using the Łojasiewicz-Simon inequality. Our approach is independent of prior work and the paper is largely self-contained.

Theorems & Definitions (184)

• Theorem 1.2: PDE-ODI principle, vague form
• Theorem 1.4: summary
• Theorem 1.5: Theorem \ref{['Thm_bowl_unique']}
• Theorem 1.6: Proposition \ref{['Prop_same_Q_close']}
• Theorem 1.7: Theorem \ref{['Thm_existence_oval']}
• Theorem 1.8: Proposition \ref{['Prop_Q_continuous']}\ref{['Prop_Q_continuous_a']}
• Corollary 1.9: Proposition \ref{['Prop_Q_continuous']}\ref{['Prop_Q_continuous_b']}, \ref{['Prop_Q_continuous_c']}
• Corollary 1.10
• Theorem 1.11: Corollary \ref{['Cor_eternal']}
• Theorem 1.12: Theorem \ref{['Thm_stability_necks']}